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\newcommand{\CourseName}{复变函数作业5AB}
\newcommand{\CourseStudents}{王立庆（2022 级数学与应用数学1班）}

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\date{2024年11月28日}
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\section{Series and Product Developments } 

\begin{enumerate}

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\item %%1
Let $f_n(z)=\frac{z}{2z^n+1}$. Let $\Omega_n$ be the disk $|z|<2^{-1/n}$. 
Show that $\lim\limits_{n\to\infty} f_n(z) =z$ in the disk $|z|<1$.   


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\item %%2
Suppose that $f_n(z)$ is analytic in the region $\Omega_n$, and that the sequence $\{f_n(z)\}$ converges to a limit function $f(z)$ in a region $\Omega$, uniformly on
every compact subset of $\Omega$.
Then $f(z)$ is analytic in $\Omega$. 
Moreover, $f_n'(z)$ converges uniformly to $f'(z)$ on every compact subset of $\Omega$.


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\item %%3
Using Taylor's theorem applied to a branch of $\log (1 + z/n)$,
prove that
$$
\lim\left(1 + \frac{z}{n}\right)^n = e^z
$$
uniformly on all compact sets.


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\item %%4
If $f(z)$ is analytic in the region $\Omega$, containing $z_0$, then the representation
$$
f(z) = f(z_0) + \frac{f'(z_0)}{1!}(z-z_0) + \cdots + \frac{f^{(n)}(z_0)}{n!}(z-z_0)^n + \cdots
$$
is valid in the largest open disk of center $z_0$ contained in $\Omega$.


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\item %%5
The expression
$$
\{f,z\} = \frac{f'''(z)}{f'(z)} - \frac{3}{2}\left(\frac{f''(z)}{f'(z)}\right)^{2}
$$
is called the Schwarzian derivative of $f$. If $f$ has a multiple zero or pole,
find the leading term in the Laurent development of $\{f,z\}$.
Answer: If $f(z) = a(z-z_0)^m + \cdots$, then $\{f,z\} = \frac{1}{2}(1-m^2)(z-z_0)^{-2} + \cdots$.

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\end{enumerate}

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\section{Partial Fractions and Factorization } 

\begin{enumerate}

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\item %%1
Let $\{b_\nu\}$ be a sequence of complex numbers with $\lim\limits_{\nu\to\infty} b_\nu = \infty$, and let $P_\nu(\xi)$ be polynomials without constant term. 
Then there are functions which are meromorphic in the whole plane with poles at the points $b_\nu$ and the corresponding singular parts $P_\nu(1/(z-b_\nu))$. 
Moreover, the most general meromorphic function of this kind can be written in the form
$$
f(z)=\sum\limits_\nu \left[ P_\nu\left(\frac{1}{z-b_\nu}\right)-p_\nu(z) \right] +g(z). 
$$
where the $p_\nu(z)$ are suitably chosen polynomials and $g(z)$ is analytic in the whole plane. 



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\item %%2
Prove that 
$$
\frac{\pi^2}{\sin^2\pi z} = \sum\limits_{n=-\infty}^{\infty} \frac{1}{(z-n)^2}
$$

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\item %%3
Prove that 
$$
\frac{\pi}{\sin\pi z} = \lim\limits_{m\to\infty}
\sum\limits_{n=-m}^{m} (-1)^n\frac{1}{z-n}
$$


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\item %%4
The infinite product $\prod\limits_{1}^{\infty} (1 + a_n)$ with $1 + a_n \neq 0$  converges simultaneously with the series $\sum\limits_{1}^{\infty}\log (1 + a_n)$ whose terms represent the values of the principal branch of the logarithm.


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\item %%5
There exists an entire function with arbitrarily prescribed zeros $a_n$ provided that, in the case of infinitely many zeros, $a_n\to\infty$. 
Every entire function with these and no other zeros can be written in the form
$$
f(z) = z^me^{g(z)}\prod\limits_{n=1}^{\infty}\left(1-\frac{z}{a_n}\right)
\exp\left[ \frac{z}{a_n} + \frac{1}{2}\left(\frac{z}{a_n}\right)^2+\cdots+\frac{1}{m_n}\left(\frac{z}{a_n}\right)^{m_n} \right]
$$
where the product is taken over all $a_n\neq 0$, the $m_n$ are certain integers, and
$g(z)$ is an entire function.


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\item %%6
Prove that 
$$
\sin \pi z = \pi z \prod\limits_{n=1}^{\infty} \left( 1-\frac{z^2}{n^2} \right).  
$$

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\item %%7
Derive the following expression of the Gamma function, 
$$
\Gamma(z) = \frac{e^{-\gamma z}}{z}\prod\limits_{n=1}^{\infty} \left( 1+\frac{z}{n} \right)^{-1}e^{z/n},
$$
where 
$$
\gamma = \lim\limits_{n\to\infty} \left( 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}-\log n \right). 
$$


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\end{enumerate}

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